Lemma 62.7.5. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a relative $r$-cycle on $X/S$. Let $\{ g_ i : S_ i \to S\} $ be an fppf covering. Then $\alpha $ is equidimensional if and only if each base change $g_ i^*\alpha $ is equidimensional.
Proof. If $\alpha $ is equidimensional, then each $g_ i^*\alpha $ is too by Lemma 62.7.3. Assume each $g_ i^*\alpha $ is equidimensional. Denote $W$ the closure of $\text{Supp}(\alpha )$ in $X$. Since $g_ i : S_ i \to S$ is universally open (being flat and locally of finite presentation), so is the morphism $f_ i : X_ i = S_ i \times _ S X \to X$. Denote $\alpha _ i = g_ i^*\alpha $. We have $\text{Supp}(\alpha _ i) = f_ i^{-1}(\text{Supp}(\alpha ))$ by Lemma 62.5.7. Since $f_ i$ is open, we see that $W_ i = f_ i^{-1}(W)$ is the closure of $\text{Supp}(\alpha _ i)$. Hence by assumption the morphism $W_ i \to S_ i$ has relative dimension $\leq r$. By Morphisms, Lemma 29.28.3 (and the fact that the morphisms $S_ i \to S$ are jointly surjective) we conclude that $W \to S$ has relative dimension $\leq r$. $\square$
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